MEDB 5501, Module12

Topics to be covered

What you will learn
- Two factor analysis of variance
- Relationship to linear regression
- Checking assumptions
- R code for two factor analysis of variance
- Interactions
- R code for interactions
- Your homework

Two factor analysis of variance

Continuous outcome
Two categorical predictors
Example
- Hearing test (decibels at high frequency)
- Age group (Old or Young)
- Gender (Female or Male)

Balanced data

Proportional number in each category level combination group
- 3 old females, 3 old males, 3 young females, 3 young males
- 6 old females, 6 old males, 2 young females, 2 young males

Unbalanced data

Unequal numbers in some category combinations
- 3 old females, 3 old males, 3 young females, 2 young males
Extreme case: empty category combinations
- 3 old females, 3 old males, 3 young females, 0 young males

Historical perspective on unbalanced data, 1

Historical perspective on unbalanced data, 2

Historical perspective on unbalanced data, 3

Book cover of Analysis of Messy Data, Volume 1

Together, they wrote a series of books with the provocative title, Analysis of Messy Data. This is the first volume in the series, published in 1984. The book deals largely with unbalanced data in analysis of variance. I must admire the bravery of writing this book. At a time when I avoided the complexities of unbalanced data in my teaching, Milliken and Johnson dove straight in.

I’ve met both Milliken and Johnson at several “Agriculture and Statistics” conferences held for many years at Kansas State University. I’ve always been more of the health care side than the agricultural side of statistics, but there is a lot that I have learned from these conferences. An interesting sidelight is that the conference always held line dancing lessons on the Sunday before the conference start. I and other geeky statisticians would get up and awkwardly follow the instructions. I never got very good at line dancing, but it was always a lot of fun.

Historical perspective on unbalanced data, 4

A simple illustration of the complexities of unbalanced data, 1

A simple illustration of the complexities of unbalanced data, 2

A simple illustration of the complexities of unbalanced data, 3

Mathematical model, 1

\(Y_{ijk}\)
- i = which level of first category
- j = which level of second category
- k = which patient within a category combination

Mathematical model, 2

\(\begin{smallmatrix} Age & Gender & Outcome \\ Old & Female & Y_{111} \\ Old & Female & Y_{112} \\ Old & Female & Y_{113} \\ Old & Male & Y_{121} \\ Old & Male & Y_{122} \\ Old & Male & Y_{123} \\ Young & Female & Y_{211} \\ Young & Female & Y_{212} \\ Young & Female & Y_{213} \\ Young & Male & Y_{221} \\ Young & Male & Y_{222} \\ Young & Male & Y_{223}\end{smallmatrix}\)

Mathematical model, 3

\(Y_{ijk} = \mu + \alpha_i + \beta_j +\epsilon_{ijk}\)
- \(i=1,...,a,\ j=1,...,b\)
- \(\Sigma \alpha_i=0,\ \Sigma \beta_j=0\)
- \(\epsilon_{ijk}\) is \(N(0,\sigma)\)
\(\bar{Y}_{i..}\) is the average for the ith level of first factor
\(\bar{Y}_{.j.}\) is the average for the jth level of second factor
\(\bar{Y}_{...}\) is the average for all of the data

Mathematical model, 4

\(SS(Total)=\Sigma_i \Sigma_j \Sigma_k\ (Y_{ijk}-\bar{Y}_{...})^2\)
- df=abn-1
\(SS(A)=\Sigma_i\ bn(\bar{Y}_{i..}-\bar{Y}_{...})^2\)
- df=a-1
\(SS(B)=\Sigma_j\ an(\bar{Y}_{.j.}-\bar{Y}_{...})^2\)
- df=b-1
\(SS(Error)=SS(Total)-SS(A)-SS(B)\)
- df=(abn-1)-(a-1)-(b-1)

Artificial data

# A tibble: 12 × 5
      id age   gender code     db
   <int> <chr> <chr>  <chr> <dbl>
 1     1 old   female of       45
 2     2 old   female of       60
 3     3 old   female of       60
 4     4 old   male   om       65
 5     5 old   male   om       60
 6     6 old   male   om       70
 7     7 young female yf       20
 8     8 young female yf       20
 9     9 young female yf        5
10    10 young male   ym       25
11    11 young male   ym       20
12    12 young male   ym       30

Artificial data with means

# A tibble: 12 × 8
      id age   gender code     db age_mean gender_mean overall_mean
   <int> <chr> <chr>  <chr> <dbl>    <dbl>       <dbl>        <dbl>
 1     1 old   female of       45       60          35           40
 2     2 old   female of       60       60          35           40
 3     3 old   female of       60       60          35           40
 4     4 old   male   om       65       60          45           40
 5     5 old   male   om       60       60          45           40
 6     6 old   male   om       70       60          45           40
 7     7 young female yf       20       20          35           40
 8     8 young female yf       20       20          35           40
 9     9 young female yf        5       20          35           40
10    10 young male   ym       25       20          45           40
11    11 young male   ym       20       20          45           40
12    12 young male   ym       30       20          45           40

SS(Total)

SS(Age)

SS(Gender)

Analysis of variance table

Analysis of Variance Table

Response: db
          Df Sum Sq Mean Sq F value    Pr(>F)    
age        1   4800  4800.0  108.00 2.595e-06 ***
gender     1    300   300.0    6.75   0.02883 *  
Residuals  9    400    44.4                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Break #1

What you have learned
- Two factor analysis of variance
What’s coming next
- Relationship to linear regression

Create indicator variables

# A tibble: 12 × 6
   age   gender code  i_young i_male    db
   <chr> <chr>  <chr>   <dbl>  <dbl> <dbl>
 1 old   female of          0      0    45
 2 old   female of          0      0    60
 3 old   female of          0      0    60
 4 old   male   om          0      1    65
 5 old   male   om          0      1    60
 6 old   male   om          0      1    70
 7 young female yf          1      0    20
 8 young female yf          1      0    20
 9 young female yf          1      0     5
10 young male   ym          1      1    25
11 young male   ym          1      1    20
12 young male   ym          1      1    30

Two factor analysis of variance using aov

m1 <- aov(db ~ age + gender, data=hearing)
anova(m1)

Analysis of Variance Table

Response: db
          Df Sum Sq Mean Sq F value    Pr(>F)    
age        1   4800  4800.0  108.00 2.595e-06 ***
gender     1    300   300.0    6.75   0.02883 *  
Residuals  9    400    44.4                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Two factor analysis of variance using linear regression, 1

m2 <- lm(db ~ age + gender, data=hearing)
anova(m2)

Analysis of Variance Table

Response: db
          Df Sum Sq Mean Sq F value    Pr(>F)    
age        1   4800  4800.0  108.00 2.595e-06 ***
gender     1    300   300.0    6.75   0.02883 *  
Residuals  9    400    44.4                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Two factor analysis of variance using linear regression, 2

tidy(m2)

# A tibble: 3 × 5
  term        estimate std.error statistic      p.value
  <chr>          <dbl>     <dbl>     <dbl>        <dbl>
1 (Intercept)     55        3.33     16.5  0.0000000492
2 ageyoung       -40        3.85    -10.4  0.00000260  
3 gendermale      10.0      3.85      2.60 0.0288

Break #2

What you have learned
- Relationship to linear regression
What’s coming next
- Checking assumptions

Checking assumptions

Normality (Non-normality)
Homogeneity (Heterogeneity)
Independence (Lack of independence)

Use the boxplot to check assumptions

Non-normality if boxplot shows skewness and/or outliers
Heterogeneity if boxplot shows large change in variation
Draw clustered boxplot to examine every combination of categories
- Use a\(\times\)b boxplots
Independence is checked qualitatively

Alternatives if assumptions violated

There is no analog to Kruskal-Wallis or Mann-Whitney-Wilcoxon
Consider a log transformation
- All values greater than 0
- Groups with larger means have larger variation
- Data is skewed right and outliers only on the high end

Break #3

What you have learned
- Checking assumptions
What’s coming next
- R code for two factor analysis of variance

Analysis of fruitfly data

Find the file simon-5501-12-moon.qmd on my github site.

Break #4

What you have learned
- R code for two factor analysis of variance
What’s coming next
- Interactions

What is an interaction

Impact of one variable is influenced by a second variable
Example, influence of alcohol on sleeping pills
Three types of interactions
- Between two categorical predictors
- Between a categorical and a continuous predictor
- Between two continuous predictors
Interactions greatly complicate interpretation

Interaction plot

X axis, first categorical variable
Separate lines for second categorical variable
Y axis, average outcome

Hypothetical interaction plots, 1

No interaction
Ineffective treatment
Boys/girls similar

No interaction
Ineffective treatment
Boys fare better than girls

An interaction plot shows the mean values for each of the two categories. In this example, there is a placebo and a treatment. The outcome is unspecified, but a larger value is presumed to represent a better outcome. This is a pediatric example and the data is subdivided into two populations, boys and girls.

The flatness or steepness of the lines indicates whether patients given the treatment fare better than patients given the placebo.

The separation (if there is any) between the lines measures whether boys fare better or worse than girls.

If the lines have roughly the same slope (both are flat or both are steep), then there is no interaction.

In the plot on the left, the two lines are flat, indicating that the treatment is ineffective. The outcome is not changed from the placebo.

The two lines lie more or less on top of one another. This indicates that there is no difference in average outcome between boys are girls.

In the plot on the right, the two lines are flat. The treatment is ineffective. There is, however, a difference. The average outcome for boys is a lot better both in the placebo group and the treatment group. The lines are roughly parallel, indicating no interaction.

Hypothetical interaction plots, 2

No interaction
Effective treatment
Boys/girls similar

No interaction
Effective treatment
Boys fare better than girls

Hypothetical interaction plots, 3

Significant interaction
Harmful treatment in boys
Effective treatment in girls

Significant interaction
Ineffective treatment in boys
Effective treatment in girls

Hypothetical interaction plots, 4

Significant interaction
Girls fare better overall
Effective treatment
Much more effective in boys

Indicator variable for an interaction

# A tibble: 12 × 7
   age   gender code  i_young i_male i_m_by_y    db
   <chr> <chr>  <chr>   <dbl>  <dbl>    <dbl> <dbl>
 1 old   female of          0      0        0    45
 2 old   female of          0      0        0    60
 3 old   female of          0      0        0    60
 4 old   male   om          0      1        0    65
 5 old   male   om          0      1        0    60
 6 old   male   om          0      1        0    70
 7 young female yf          1      0        0    20
 8 young female yf          1      0        0    20
 9 young female yf          1      0        0     5
10 young male   ym          1      1        1    25
11 young male   ym          1      1        1    20
12 young male   ym          1      1        1    30

Interpretation of intercept and slopes

When you can’t estimate an interaction

Special case, n=1
- Only one observation for categorical combination

Example, full moon study, 1 of 2

# A tibble: 36 × 3
   month1 moon1      n
   <fct>  <fct>  <int>
 1 Aug    Before     1
 2 Aug    During     1
 3 Aug    After      1
 4 Sep    Before     1
 5 Sep    During     1
 6 Sep    After      1
 7 Oct    Before     1
 8 Oct    During     1
 9 Oct    After      1
10 Nov    Before     1
# ℹ 26 more rows

Example, full moon study, 2 of 2

m1 <- aov(admission ~ month*moon, data=er)
anova(m1)

Analysis of Variance Table

Response: admission
           Df Sum Sq Mean Sq F value Pr(>F)
month      11 455.58  41.417     NaN    NaN
moon        2  41.51  20.757     NaN    NaN
month:moon 22 127.82   5.810     NaN    NaN
Residuals   0   0.00     NaN

Break #5

What you have learned
- Interactions
What’s coming next
- R code for interactions

Analysis of fruitfly data

Find the file simon-5501-12-fruitfly.qmd on my github site.

Break #6

What you have learned
- R code for interactions
What’s coming next
- Your homework

Your homework

Find the file simon-5501-12-directions.md on my github site.

Summary

What you have learned
- Two factor analysis of variance
- Relationship to linear regression
- Checking assumptions
- R code for two factor analysis of variance
- Interactions
- R code for interactions
- Your homework